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SithsAndGiggles
 one year ago
Best ResponseYou've already chosen the best response.2Recall the distance formula: \[d^2=(xx_0)^2+(yy_0)^2\] In this particular problem, you have \((x_0,y_0)=(0,0)\) (the origin), since you want to find the distance between any given point on the curve \((x,y)\) and the origin. \[d^2=x^2+y^2\] Substitute the given equation of the curve: \[d^2=x^2+\left(4\sqrt{2x+2}\right)^2\\ d^2=x^2+16(2x+2)\\ d^2=x^2+32x+32\] Now differentiate both sides implicitly with respect to some dummy variable \(t\): \[2d\frac{dd}{dt}=(2x+32)\frac{dx}{dt}\] You're told that at the point \((1,8)\), \(x\) is increasing at a rate of 3 units/s, which translates to \(\dfrac{dx}{dt}=3\). \[2d\frac{dd}{dt}=(2\cdot1+32)(3)\\ 2d\frac{dd}{dt}=102\] You're asked to find \(\dfrac{dd}{dt}\), so first you need to find \(d\). To do this, find the distance between \((1,8)\) and \((0,0)\), then plug it into the above equation. Lastly, solve for \(\dfrac{dd}{dt}\).

mathcalculus
 one year ago
Best ResponseYou've already chosen the best response.0@SithsAndGiggles omg thank you!! :)

mathcalculus
 one year ago
Best ResponseYou've already chosen the best response.0@SithsAndGiggles hey, i found the distance which is 8 right? then i plugged in 8 into the equation and got 51/8... however, something is wrong?

satellite73
 one year ago
Best ResponseYou've already chosen the best response.0i get \(\frac{19}{8}\)

mathcalculus
 one year ago
Best ResponseYou've already chosen the best response.0to find the distance is this:

mathcalculus
 one year ago
Best ResponseYou've already chosen the best response.0i got it... thank you! @satellite73

satellite73
 one year ago
Best ResponseYou've already chosen the best response.0\[d^2=x^2+32x+32\] \[2dd'=2xx'+32x'\] \[dd'=xx'+32x'\] \[8d'=3+96\]

satellite73
 one year ago
Best ResponseYou've already chosen the best response.0distance isn't 8 is it? distance is \(\sqrt{65}\)

satellite73
 one year ago
Best ResponseYou've already chosen the best response.0which is pretty close to 8...

satellite73
 one year ago
Best ResponseYou've already chosen the best response.0is the final answer \(\frac{99}{\sqrt{65}}\) ?

mathcalculus
 one year ago
Best ResponseYou've already chosen the best response.0close yes... i thought i had to do this: x2x1/y2y1

mathcalculus
 one year ago
Best ResponseYou've already chosen the best response.0it's 51/sqrt(65)/65

satellite73
 one year ago
Best ResponseYou've already chosen the best response.0oh damn i forgot to divide 32 by 2 must be past my bedtime
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